• You are attempting to read (while(<FILE>){) from your output file. You'll never reach the print.
• push @new, $i; adds undef for the lines grep matched.
• ($i)= grep { m/^#/ } $_; keeps the comment lines instead of the lines without comments.
• for(@array){ ($i)= grep { m/^#/ } $_;  push @new, $i; }
  should be
  @new = grep { !m/^#/ } @array;
• You don't check if your open succeeded
  open(FILE, "<$file") or die("Unable to open input file \"$file\": $!\n");
  open(FILE, ">$nfile") or die("Unable to create output file \"$nfile\": $!\n");
• You should use the 3-arg open whenever possible.
  open(FILE, '<', $file)
  open(FILE, '>', $nfile)
• You should use lexicals whenever possible.
  open(my $fh_in, '<', $file) (while (<$fh_in>))
  open(my $fh_out, '>', $nfile) (print $fh_out ...)
• Shouldn't $file=<>; be $file=@ARGV[0];?
  Shouldn't $nfile=<>; be $nfile=@ARGV[1];?
  Parameters are usually passed as parameters instead of via STDIN. If you do mean to read from STDIN, give the user a prompt, and use <STDIN> instead of <>.
• No checking to see if $file and $nfile were specified. (They could be undef.)
• The trailing newline is not removed from $file and $nfile.
• Why read the whole file into memory?

Fix: (one-liner) ( Added missing >. Thanks Hofmator. )

perl -ne "print unless /^#/" infile > outfile
[download]

Fix: (program -- memory efficient)

use strict;
use warnings;

die("usage: $0 {infile} {outfile}\n";
   if @ARGV != 2;

my ($file_in, $file_out) = @ARGV;

open(my $fh_in, '<', $file_in)
   or die("Unable to open input file \"$file_in\": $!\n");

open(my $fh_out, '>', $file_out)
   or die("Unable to create output file \"$file_out\": $!\n");

while (<$fh_in>) {
   if (!/^#/) {
      print $fh_out $_;
   }
}
[download]

Fix: (program -- (more) compact)

use strict;
use warnings;

die("usage: $0 {filein} {fileout}\n";
   if @ARGV != 2;

my ($file_in, $file_out) = @ARGV;

open(my $fh_in, '<', $file_in)
   or die("Unable to open input file \"$file_in\": $!\n");

open(my $fh_out, '>', $file_out)
   or die("Unable to create output file \"$file_out\": $!\n");

print $fh_out (grep !/^#/, <$fh_in>);
[download]